Nyquist Frequency in Sequentially Sampled Data

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Munich Personal RePEc Archive

Nyquist Frequency in Sequentially

Sampled Data

Faghih, Nezameddin and Faghih, Ali

Shiraz University, University of Maryland

2008

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Nyquist Frequency in Sequentially Sampled Data

Nezameddin Faghih* and Ali Faghih**

ABSTRACT

This paper studies the sequential sampling scheme as a solution to the problem of aliasing, where the sampling interval is restricted to a minimum allowable value T. In sequential sampling, the signal is sampled at intervals of T, T+, T+2, T+3, ...; where   T and  may be selected as desirable. Sequential sampling is, however, analyzed and it is proved that when the ratio T/ is an integral number, the associated spectral estimates give a Nyquist frequency 1

2. This sampling scheme can, therefore, be employed to yield a required cut- off frequency.

* Shiraz University ([email protected])

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2 INTRODUCTION

Some data acquisition systems have a minimum allowable sampling interval and do not provide a desired sampling period less than a minimum allowable value. This may be due to some restrictions set by the measuring instrument that has to be used [1-7].

Let the minimum allowable sampling time be T; if the uniform sampling scheme is employed then, the Nyquist or cut-off frequency is known [8] to be given as:

f

T

c 

1

2 (1)

This would mean that if frequencies higher than fc are present,

aliasing will occur. Otherwise, the signal would have to be filtered so that only frequencies below fc are passed and, therefore, the spectral analysis

will be restricted [8].

The sequential sampling scheme can, however, be employed to obtain an autocorrelation function with estimates  apart, where  T, with the exception of coefficients lying inside the range R(0)R(T). In this sampling scheme, the signal would be sampled at intervals of:

T, T+, T+2, T+3, ...

From the sampled signal, an autocorrelation function can be obtained with the coefficients:

R(0), R(T), R(T+), R(T+2), ...

While T is restricted, the value of  may be chosen as desirable. It will be proved, in this paper, that the sequential sampling can give an increased cut-off frequency as:

f_cs 1

2 (2)

The sequential sampling can, therefore, be employed to overcome aliasing and the restrictions of spectral analysis, by selecting a sufficiently small value for .

THE CUT OFF FREQUENCY IN THE SEQUENTIAL

SAMPLING

The cut- off frequency provided by the sequential sampling scheme is considered in this section. The analysis employs the impulse representation of a continuous signal as an approach to discretization [9-14].

In the sequential sampling, the signal is sampled at intervals of:

T, T+, T+2, T+3, ... The sampling instants are, therefore, given by:

ti= 0, T, 2T+, 3T+3, 4T+6, ... (3)

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t_i i T r i i

r i

   





 [ ], 0 1 2 3, , , ,...

0

(4)

Since, r i i

r i

 





2 1

0

( ) , then equation (4) gives:

t_i i T  i i 

2( 1) (5)

When a continuous signal x(t) is sampled, the sample values x(ti) are

acquired. A discrete autocorrelation function, with coefficients R(j), may

be obtained from the discrete signal, by contributions of the products x(ti)x(ti+j). Equation (5) can be used to give the time delay j as:

_jt_{i j}_ t_i T[ (j j   ) ij (j ) ] 

2 1 1 (6)

i= o, 1, 2, ...

j= o, 1, 2, ...

where  is a constant given by:

T/ (7)

It is seen from equation (6) that for j=0, the time delay is zero and for j=1, the time delay is T+i (where i=0, 1, 2,3, ...). An autocorrelation function is, therefore, obtainable at discrete values of the time delay given as:

n =o, T+n n= 0, 1, 2, 3, ... (8)

If the ratio  is an integral number, then higher values of j would also provide more contributions to the autocorrelation estimates at the above time delays n. This is because j(j-1)/2 is always even, and any value of j

would hence add a multiple of  to T.

The discrete autocorrelation function may be represented as:

R*( )  . ( )R   _b( ) (9)

where R() is the continuous autocorrelation function and b() is the

following form of the delta comb:

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4

It is established [9-10] that the Fourier transform of equation (10) can be written as:

_b_{( )}_ _{ }₁ _ejT

_

_ej n  0

(11)

which by manipulation [9-10] can be re-written as:

_b j j

j

e e

e ( )

 



   

 



1

(12)

where substitution has also been made for T from equation (7).

The Fourier transformation of R*() gives the spectral density S*() corresponding to the sampled signal and that of R() would yield the spectral density S() of the original continuous signal. The approach adopted for the Fourier transformation of equation (9) is based on the convolution and residue theorems [9]. By evaluating the residue terms [9] and using the convolution property [9], for substitution into equation (9), the Fourier transform of this equation ca be obtained as:

S* ( )  ej  n S( n_cs) , _cs_/

 



2 ₂

 (13)

However, if the ratio =(T/) is an integral number then, ej2 n 1

noting that n is also an integer. Substituting this into equation (13) gives:

S* ( )  S( n_cs) , _cs_/

 



2 _ (14)

Now, consider the periodicity of S*(); this can also be examined by applying the corresponding methods [9-14]. Using equations (9), (10), (11) and the rules established for discrete Fourier transformation [9-10], it can be written:

[ ( )R0 e j T R T n( )e j n] [ ( )R0 e j R( n )e j n]

0 0

 



     



   (15)

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                 [ ( ) ( ) ] [ ( ) ( ) . ] [ ( ) . ( ) . ] [ ( ) . ( ) ] ( ) ( )

R e R T n e

R e e R n e e

R e e R n e

j m j m n

j j m j n j m n

j j m j n j mn

j j m j n

cs cs cs cs 0 0 0 0 2 2 0 2 2 0 2 2 0 2 0                              



                

since m and n are integers. If  is also an integral number, this would reduce to:

[ ( )R 0 e j R(T n)e j n]

0

  



   

from which it follows that:

S*(2m_cs)S*( ) (17) This is the mathematical statement for S*() to be periodic with period 2cs. Otherwise, if  is not an integral number,

S*(2m_cs)S*( ) (18) and the requirement for periodicity is not met.

It is, therefore, seen that when the ratio  is an integral number, the periodic pattern, conforming with the Nyquist theorem,[9-14] is obtained. That is, the sequential sampling gives a cut-off frequency

_cs _/_ or f_cs  1

2. On the contrary, when  is not a whole number, S * ( ) is related to the true spectral density by equation (13); it includes a complex term and is not periodic.

CONCLUSIONS

This paper has considered the sequential sampling scheme, as a solution to the problem of aliasing, where the sampling interval is restricted to a minimum allowable valueT . In the sequential sampling, the signal is sampled at intervals of   T,  , 2 , 3,...; where

 p and may be selected as desirable.

The sequential sampling was considered analytically and it was proved that, when the ratio  / is an integral number, the corresponding spectral estimates give a cut- off frequency of 1

2. On the contrary, when the ratio is not a whole number, the associated spectrum of the sequentially

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6

sampled data was found to comprise a complex term in its relation to the true spectrum and would not be periodic in terms of the cut-off frequency.

.

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